1.13 Problems
Section 1.2: Sample Space and Events
1.1 A fair die is rolled twice. Find the probability of the following events:
a. The second number is twice the first. 3/6
b. The second number is not greater than the first. 7/12
c. At least one number is greater than 3. 27/36
1.2 Two distinct dice A and B are rolled. What is the probability of each of the
following events?
a. At least one 4 appears. 11/36
b. Just one 4 appears. 10/36
c. The sum of the face values is 7. 6/36
d. One of the values is 3 and the sum of the two values is 5.2/36
e. One of the values is 3 or the sum of the two values is 5.11/36 +4/36 =15/36
1.3 Consider an experiment that consists of rolling a die twice.
a. Plot the sample space S of the experiment. S = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6),
(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), (4, 1),
(4, 2), (4, 3), (4, 4), (4, 5), (4, 6), (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), (6, 1), (6, 2),
(6, 3), (6, 4), (6, 5), (6, 6)}
b. Identify the event A, which is the event that the sum of the two outcomes
is equal to 6. S = { (1, 5), (2, 4), (3, 3), (4, 2), (5, 1)}
c. Identify the event B, which is the event that the difference between the
two outcomes is equal to 2.
{(1, 3),(2, 4),(3, 1),(4, 2),(4, 6), (6, 4),(3, 5),(5, 3)}
1.4 Afour-sided fair die is rolled twice.What is the probability that the outcome
of the first roll is greater than the outcome of the second roll?15/36 =5/12
1.5 A coin is tossed until the first head appears, and then the experiment is
stopped. Define a sample space for the
experiment.{H,TH,TTH,TTTH,TTTTH,TTTTTH,TTTTTTH,TTTTTTTH,………………‫}غير منتهية‬
1.6 A coin is tossed four times and observed to be either a head or a tail each
time. Describe the sample space for the experiment.
{(H,H,H,H) , (H,H,H,T), (H,H,T,H) , (H,T,H,H) , (T,H,H,H) , (T,T,T,T) , (T,T,T,H) , (T,T,H,T) ,
(T,H,T,T) , (H,T,T,T) , (H,H,T,T) , (T,T,H,H) , (H,T,H,T) , (T,H,T,H) , (H,T,T,H) , (T,H,H,T) }
48 Chapter 1 Basic Probability Concepts
1.7 Three friends, Bob, Chuck, and Dan take turns (in that order) throwing a
die until the first “six” appears. The person that throws the first six wins the
game, and the game ends.Write down a sample space for this game.
‫ العبين هما بوب وشاك ودان كل واحد يرمي الزهرة بالدور لحد ما يظهر لواحد فيهم الرقم‬3 ‫المسألة إنو في‬
‫ وطلب في السؤال كتابة السامبل سبيس‬, ‫ ألي واحد فيهم يعتبر هو الفائر وتنتهي اللعبة‬6 ‫ أول ما يظهر‬6
‫لهذه التجربة‬
‫ أو‬Dan ‫ أو للثالث‬Chuck‫ أو للتاني‬Bob‫ حتظهر إما لالعب األول‬6 ‫إذن بما إنهم بيلعبوا بالترتيب إذن‬
‫ لواحد فيهم إذن السابل سبيس حتكون‬6 ‫إنهم حيعيدوا اللفة من جديد إلين ما يظهر الرقم‬
{( Bob) , (Chuck) ,( Dan) , ( Bob Chuck) , (Bob Chuck Dan) , (Bob Chuck Dan
Bob) , (Bob Chuck Dan Bob Chuck) ,( Bob Chuck Dan Bob Chuck Dan)
……………………….‫}غير منتهية‬
Section 1.3: Definitions of Probability
1.8 A small country has a population of 17 million people of whom 8.4 million
aremale and 8.6million are female. If 75% of themale population and 63%
of the female population are literate, what percentage of the total population
is literate?
^ . ^ ‫هادي المسألة ححلها بالرسم الشجري عشان تكون واضحة سوري يا جماعة الرسمة شوية معوقة‬
literate p(L)=0,75
Mail p(m)=8,4/17
Not literate p(not L)=1-0,75=0,25
Femail p(f) =8,6/17
literate p(L)=0.63
not literate p(not L)= 0,37
percentage of the total population is literate = P(m∩L) ∪ p(f∩L)
8.4
=
17
× 0.75 +
8.6
17
× 0.63 = 0.69 ≫ 69%
1.9 Let A and B be two independent events with P[A] = 0.4 and P[A∪B] = 0.7.
What is P[B]?
‫أوالً نكتب القانون‬P(A ∪ B) = P(A) +P(B) −P(A ∩ B)
‫بالتعويض‬
0.7
= 0.4+
P(B) - P(A× 𝐵)
0.7
= 0.4+
P(B) - P(A) × 𝑝(𝐵)
0.7
= 0.4+
P(B) - 0.4× 𝑝(𝐵)
‫ وتحل كأي معادلة عادية ذات مجهول‬p(B) ‫اآلن المعادلة صار فيها مجهول واحد ال هو‬
)‫واحد (كملوها إنتو يا كتاكيت‬
1.10 Consider two events A and B with known probabilities P[A], P[B], and
P[A ∩ B]. Find the expression for the event that exactly one of the two
events occurs in terms of P[A], P[B], and P[A ∩ B].
‫هذا السؤال ماااااااااالو دااااااعي فلسفة على الفاضي أحذفوا<<<<هيي هيي أشوفك تحذفي على‬
‫كيفك<<< حوديكم في داهية هههههههه أمزح ) األفضل تسألو األستاذة عنو‬
1.11 Two events A and B have the following probabilities: P[A] = 1/4,
P[B|A] = 1/2, and P[A|B] = 1/3. Compute
(a) P[A ∩ B] =1/8
, (b) P[B] = 3/8
(c) P[A ∪ B]. P(A ∪ B) = P(A) +P(B) −P(A ∩ B)
1
4
+
3
8
−
1
8
1
=2
1.12 Two events A and B have the following probabilities: P[A] = 0.6,
P[B] = 0.7, and P[A ∩ B] = p. Find the range of values that p can take.
‫هذا السؤال معقد شوية وأستاذتنا حذفتو أسألوا للتأكيد‬
1.13 Two events A and B have the following probabilities: P[A] = 0.5,
P[B] = 0.6, and P[A ∩ B] = 0.25. Find the value of P[A ∩ B].
P(A ∪ B) = P(A) +P(B) −P(A ∩ B)
‫سهل‬
1.14 Two events A and B have the following probabilities: P[A] = 0.4,
P[B] = 0.5, and P[A ∩ B] = 0.3. Calculate the following:
a. P[A ∪ B]= 0.6
b. P[A ∩ B]=0.1
c. P[A ∪ B]=0.7
1.15 Christie is taking a multiple-choice test in which each question has four
possible answers. She knows the answers to 40% of the questions and can
narrow the choices down to two answers 40% of the time. If she knows
nothing about the remaining 20% of the questions, what is the probability
that she will correctly answer a question chosen at random from the test?
)‫<<<<<<<شغاالة حذف عجبتني الحكاية (صراحة األستاذة قالت مو مهم‬
‫هذا السؤااال أحذفووووووووووووووووووووه‬
1.16 A box contains nine red balls, six white balls, and five blue balls. If three
balls are drawn successively from the box, determine the following:
a. The probability that they are drawn in the order red, white, and blue if
each ball is replaced after it has been drawn.
9
20
×
6
20
×
5
20
b. The probability that they are drawn in the order red, white, and blue if
each ball is not replaced after it has been drawn.
9
20
6
5
× 19 × 18
1.17 Let A be the set of positive even integers, let B be the set of positive integers
that are divisible by 3, and let C be the set of positive odd integers. Describe
the following events:
‫ أنا ححل أول فقرة وإنتو كملوا‬, ‫هذا السؤال سهل جدا يعتمد على الفهم والجواب حيكون شرح كالمي‬
a. E1 = A∪ B = the set
of even integers and positive integers that are divisible by 3
b. E2 = A∩ B
c. E3 = A∩ C
d. E4 = (A∪ B) ∩ C
e. E5 = A∪ (B ∩ C)
1.18 A box contains four red balls labeled R1, R2, R3, and R4; and three white
balls labeled W1, W2, and W3. A random experiment consists of drawing a
ball from the box. State the outcomes of the following events:
a. E1, the event that the number on the ball (i.e., the subscript of the ball) is even.{ R2, R4, W2}
b. E2, the event that the color of the ball is red and its number is greater than 2.{ R3, R4}
c. E3, the event that the number on the ball is less than 3.{ R1, R2, W1, W2}
d. E4 = E1 ∪ E3.{ R2, W2}
e. E5 = E1 ∪ (E2 ∩ E3). .{ R2, R4, W2}
1.19 A box contains 50 computer chips of which 8 are known to be bad. A chip
is selected at random and tested.
8
(a) What is the probability that it is bad?
50
(b) If a test on the first chip shows that it is bad, what is the probability that
a second chip selected at random will also be bad, assuming the tested
8
chip is not put back into the box?
50
×
7
49
(c) If the first chip tests good, what is the probability that a second chip
selected at random will be bad, assuming the tested chip is not put
back into the box?
42
50
×
8
49
Section 1.5: Elementary Set Theory
1.20 A set S has four members: A,B,C, and D. Determine all possible subsets
of S.
s={{∅}{a}{b}{c}{d}{a,b}{a,c}{a,d},{b,c}{b,d}{a,b,c,d}{d,c}{b,c,d}{a,b,d}
{a,b,c}{a,c,d}}
1.21 For three sets A,B, and C, use the Venn diagram to show the areas corresponding
to the sets (a) (A ∪ C) − C, (b) B ∩ A, (c) A ∩ B ∩ C, and
(d) (A∪ B) ∩ C.
A) ……………………………….b)………………………………….c)………………………………..d)
1.22 A universal set is given by S = {2, 4, 6, 8, 10, 12, 14}. If we define two sets
A = {2, 4, 8} and B = {4, 6, 8, 12}, determine the following: (a) A, (b) B−A,
(c) A∪ B, (d) A∩ B, (e) A∩ B, and (f) (A ∩ B) ∪ (A ∩ B).‫محذوووووووووووووووووووف‬
1.23 Consider the switching networks shown in Figure 1.17. Let Ek denote the
event that switch Sk is closed, k = 1, 2, 3, 4. Let EAB denote the event that
there is a closed path between nodes A and B. Express EAB in terms of the
Ek for each network.‫محذووووووووووووووووووووووف‬
1.24 Let A, B, and C be three events. Write out the expressions for the following
events in terms of A, B, and C using set notation:
a. A occurs but neither B nor C occurs.
b. A and B occur, but not C.
c. A or B occurs, but not C.
d. Either A occurs and not B, or B occurs and not A.
a) (c∪ B) ∩A
b)c∩(a∩b)
C)C∩(A∪B)
D)( A∩B) ∪ (B∩A)
Section 1.6: Properties of Probability
1.25 Mark and Lisa registered for Physics 101 class. Mark attends class 65% of
the time and Lisa attends class 75% of the time. Their absences are independent.
On a given day, what is the probability that
(a) at least one of them is in class?
0.75 × 0.35 + 0.65 × 0.25 + 0.75 × 0.65 = 0.912
(b) exactly one of them is in class? 0.75 × 0.35 + 0.65 × 0.25 = 0.422
(c) Mark is in class, given that only one of them is in class?0.41 ‫مش متأكدة من هذه الفقرة‬
1.26 The probability of rain on a day of the year selected at random is 0.25 in
a certain city. The local weather forecast is correct 60% of the time when
the forecast is rain and 80% of the time for other forecasts. What is the
probability that the forecast on a day selected at random is correct?
‫محذووووووووووووووووووووووووووووووووووووووووووف‬
1.27 53% of the adults in a certain city are female, and 15% of the adults are
unemployed males.
(a) What is the probability that an adult chosen at random in this city is an
employed male?0.4 × 0.85 = 0.34
(b) If the overall unemployment rate in the city is 22%, what is the probability
that a randomly selected adult is an employed female?0.53 × 0.93 = 0.49
1.28 A survey of 100 companies shows that 75 of them have installed wireless
local area networks (WLANs) on their premises. If three of these companies
are chosen at random without replacement, what is the probability that
each of the three has installed WLANs?
75∁3
= 0.417
100∁3
Section 1.7: Conditional Probability
1.29 A certain manufacturer produces cars at two factories labeled A and B.
Ten percent of the cars produced at factory A are found to be defective,
while 5% of the cars produced at factory B are defective. If factory A produces
100,000 cars per year and factory B produces 50,000 cars per year,
compute the following:
(a) The probability of purchasing a defective car from the manufacturer? 0.083
(b) If a car purchased from the manufacturer is defective, what is the probability
that it came from factory A? 0.8
1.30 Kevin rolls two dice and tells you that there is at least one 6. What is the
probability that the sum is at least 9?
7
11
1.31 Chuck is a fool with probability 0.6, a thief with probability 0.7, and neither
with probability 0.25.
(a) What is the probability that he is a fool or a thief but not both?0.55
(b) What is the conditional probability that he is a thief, given that he is
not a fool? ‫حل هذي الفقرة شوية طويل‬
1.32 Studies indicate that the probability that a married man votes is 0.45, the
probability that a married woman votes is 0.40, and the probability that a
married woman votes given that her husband does is 0.60. Compute the
following probabilities:
(a) Both a man and his wife vote? 0.27
𝟎.𝟐𝟕
(b) A man votes given that his wife does.
𝟎.𝟒
1.33 Tom is planning to pick up a friend at the airport. He has figured out that
the plane is late 80% of the time when it rains, but only 30% of the time
when it does not rain. If the weather forecast that morning calls for a 40%
chance of rain, what is the probability that the plane will be late? 0.5
1.34 ‫محذووووووووووووووووووووووف على ضمانتي‬
1.35 A group of students consists of 60% men and 40% women. Among the
men, 30% are foreign students, and among the women, 20% are foreign
students, A student is randomly selected from the group and found to be a
foreign student.What is the probability that the student is a woman?0.376
1.36 ‫محذوووووووووووووووووووووووووووووووووووووف‬
1.37 Three car brands A, B, and C, have all the market share in a certain city.
Brand A has 20% of the market share, brand B has 30%, and brand C has
50%. The probability that a brand A car needs a major repair during the
first year of purchase is 0.05, the probability that a brand B car needs a
major repair during the first year of purchase is 0.10, and the probability
that a brand C car needs a major repair during the first year of purchase
is 0.15.
a. What is the probability that a randomly selected car in the city needs a
major repair during its first year of purchase?0.115
b. If a car in the city needs a major repair during its first year of purchase,
what is the probability that it is a brand A car?0.087
Section 1.8: Independent Events
1.38 If I toss two coins and tell you that at least one is heads, what is the probability
1
that the first coin is heads?
3
1.39 ‫أحس إنو مالو داعي أصال ما يجي زي كدا إثباتات‬
1.40 )((‫يمكن محذوف ما أعرف)أسألوا‬
Section 1.10: Combinatorial Analysis
1.42 Four married couples bought tickets for eight seats in a row for a football
game.
a. In how many different ways can they be seated?
8!
b. In how many ways can they be seated if each couple is to sit together
with the husband to the left of his wife?
4!
c. In how many ways can they be seated if each couple is to sit together?
4!X2!
d. In how many ways can they be seated if all the men are to sit together
and all the women are to sit together?
2!X4!X4!
1.43 A committee consisting of three electrical engineers and three mechanical
engineers is to be formed from a group of seven electrical engineers and
five mechanical engineers. Find the number of ways in which this can be
done if
a. any electrical engineer and any mechanical engineer can be included.
5∁3𝑋7∁3
12∁6
b. one particular electrical engineer must be on the committee
5∁3𝑋6∁2
c. two particular mechanical engineers cannot be on the same committee.
[3∁2𝑋2∁1 + 3𝐶3]𝑋7𝐶3
1.44
)‫(من قال ال أدري فقد أفتى‬
1.45 A committee of three members is to be formed consisting of one representative
from labor, one from management, and one from the public. If there
are seven possible representatives from labor, four from management, and
five from the public, how many different committees can be formed?
7𝐶1𝑋4𝐶1𝑋5𝐶1
1.46 There are 100 U.S. senators, two from each of the 50 states.
(a) If two senators are chosen at random, what is the probability that they
are from the same state?
50𝐶1
100𝐶2
(b) If ten senators are randomly chosen to form a committee, what is the
probability that they are all from different states?
50𝐶10
100𝐶10
1.47 A committee of seven people is to be formed from a pool of 10 men and 12
women.
‫‪(a) What is the probability that the committee will consist of three men‬‬
‫‪10𝐶3𝑋12𝐶4‬‬
‫‪22𝐶7‬‬
‫‪10𝐶7‬‬
‫‪22𝐶7‬‬
‫?‪and four women‬‬
‫?‪(b) What is the probability that the committee will consist of all men‬‬
‫باقي األسئلة مو معاكم‪..................‬‬
‫تم بحمد هللا‬
‫اللهم صل على محمد وعلى آل محمد كما صليت على إبراهيم وعلى آل إبراهيم‬
‫أخواني وأخواتي ال أريد سوى دعوة بظهر الغيب أن يجعل هللا لي من أمري يسرا ً وأن يحقق‬
‫آمالي في الدنيا واآلخرة‪.‬‬